Perturbative and non-perturbative infrared behavior of ...

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6. Quantization, fixed points and RG flows spectrum of fixed points spans a seven dimensional hypersurface given by the equations |α2| 2 = |α3| 2 = |λ1| 2 = |λ2| 2 = 1 NNfK 2 1 K2 2 1 MN ′ f K2 1 K2 2 1 4(NNf + 1)K 2 1 1 4(MN ′ f + 1)K2 2 K 2 2 (2NM + NNf + 1) + K 2 ′ 1 2NM + MN f + 1 +2K1K2 (NM + 2) − 4|h2| 2 K 2 1K 2 2 (MN + 1) − K 2 1K2 2 (|h3| 2 + |h4| 2 )MN + (h3 ¯ h4 + h4 ¯ h3) + |α4| 2 MN ′ f K 2 2 (2NM + NNf + 1) + K 2 ′ 1 2NM + MN f + 1 +2K1K2 (NM + 2) − 4|h1| 2 K 2 1K 2 2 (MN + 1) (|h3| 2 + |h4| 2 )MN + (h3 ¯ h4 + h4 ¯ h3) + |α1| 2 NNf − K 2 1 K2 2 2NM + NNf + 1 − K 2 1 |λ3| 2 MN ′ f + (|α1| 2 + |α2| 2 )MN 2NM + MN ′ f + 1 − K2 2 |λ3| 2 NNf + (|α3| 2 + |α4| 2 )MN When K1 = −K2 ≡ K a particular point on this surface corresponds to h1 = h2 = 0 , h3 = −h4 = 1 K λ1 = −λ2 = 1 2K , λ3 = 0 α1 = α2 = 1 K , α3 = α4 = − 1 K and describes the N = 3 ABJ/ABJM models with flavor matter [82, 83, 84]. More generally, allowing K2 = −K1 we find the fixed point h1 = h2 = 1 1 + 2 K1 1 K2 , h3 = 1 λ1 = 1 2K1 , λ2 = 1 2K2 , λ3 = 0 α1 = α2 = 1 , α3 = α4 = 1 K1 K2 K1 , h4 = 1 K2 (6.3.53) (6.3.54) (6.3.55) which corresponds to a superconformal theory obtained from the N = 3 theory of [79] by the 102
addition of flavor matter [83]. The superpotential Spot = d 3 xd 2 1 1 θ Tr + 2 K1 1 (A1 B1) K2 2 + (A 2 B2) 2 + 1 (A K1 1 B1A 2 B2) + 1 + 1 K1 ˜Q1A i BiQ1 can be thought of as arising from the action (A K2 2 B1A 1 B2) + 1 6.3. The spectrum of fixed points (Q1 2K1 ˜ Q1) 2 + 1 (Q2 2K2 ˜ Q2) 2 + 1 ˜Q2BiA K2 i Q2 + h.c. S = SCS + Smat + d 3 xd 2 θ − K1 2 Tr(Φ21) + Tr(BiΦ1A i ) + Tr( ˜ Q1Φ1Q1) + d 3 xd 2 θ − K2 2 Tr(Φ22 ) + Tr(AiΦ2Bi) + Tr( ˜ Q2Φ2Q2) (6.3.56) + h.c. (6.3.57) after integration on the Φ1,Φ2 chiral superfields belonging to the adjoint representations of the two gauge groups and giving the N = 4 completion of the vector multiplet. Therefore, as in the unflavored case, the theory exhibits N = 3 supersymmetry with the couples (A,B † )i, (Q, ˜ Q † ) r 1 and (Q, ˜ Q † ) r′ 2 realizing (2 + Nf + N ′ f ) N = 4 hypermultiplets (The CS terms break N = 4 to N = 3). As already discussed, in the absence of flavors the N = 3 superconformal theory is connected by the line of fixed points (6.3.48) to a N = 2, SU(2)A × SU(2)B invariant theory. We now investigate whether a similar pattern arises even when flavors are present. To this end, we first choose h1 = h2 = 1 2 (h3 + h4) , α1 = α2 , α3 = α4 (6.3.58) with λj arbitrary. This describes a set of N = 2 theories with global SU(2) invariance in the bifundamental sector. Solving the equations βνj = 0 for real couplings we find a whole line of SU(2)A × SU(2)B invariant fixed points parametrized by the unconstrained coupling λ3 α1 = α3 = 0 h3 = −h4 = (2NM + MN ′ f + 1)K2 1 + 2(MN + 2)K1K2 + (2MN + NNf + 1)K 2 2 λ 2 1 = 2MN + NNf + 1 − K 2 1 MN′ f λ2 3 4K 2 1 (NNf + 1) λ 2 2 = 2MN + MN′ f + 1 − K2 2NNfλ 2 3 4K2 2 (MN′ f + 1) 2(MN − 1)K 2 1 K2 2 (6.3.59) 103
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Università degli Studi di Milano-B
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Riassunto della tesi dimensionale.
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CONTENTS 4.3.2 The superpotential p
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List of Figures 3.1 New vertices pr
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Introduction and outline The advent
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Introduction and outline nonanticom
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Part I Nonanticommutative theories
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1. Basics and motivations ingredien
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1. Basics and motivations to obtain
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1. Basics and motivations 8
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2. Nonanticommutative superspace wh
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2. Nonanticommutative superspace wh
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2. Nonanticommutative superspace op
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3. The Wess-Zumino model We stress
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3. The Wess-Zumino model Φ 2 Φ U
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3. The Wess-Zumino model dim U(1)R
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3. The Wess-Zumino model • ω2 =
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4. Gauge theories ever, a modified
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4. Gauge theories They can be expre
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4. Gauge theories of the U(1) field
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4. Gauge theories and the pure gaug
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4. Gauge theories Since for the chi
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4. Gauge theories Figure 4.1: Gauge
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4. Gauge theories 4.2.2 Three-point
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4. Gauge theories 4.2.4 (Super)gaug
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4. Gauge theories and supergauge in
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4. Gauge theories where the trace o
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4. Gauge theories Figure 4.4: One-l
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4. Gauge theories fore, one-loop re
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4. Gauge theories α ˙α Γ dim R-
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4. Gauge theories 3. Gauge sector.
Page 66 and 67: 4. Gauge theories We have introduce
Page 68 and 69: 4. Gauge theories the covariant pro
Page 70 and 71: 4. Gauge theories We note that the
Page 72 and 73: 4. Gauge theories satisfy this set
Page 74 and 75: 4. Gauge theories In order to cance
Page 76 and 77: 4. Gauge theories of [22] the most
Page 78 and 79: 4. Gauge theories By direct calcula
Page 80 and 81: 4. Gauge theories pole coefficients
Page 82 and 83: 4. Gauge theories superfields, in m
Page 84 and 85: 4. Gauge theories 70 φ h ∂ Γ (a
Page 86 and 87: 4. Gauge theories is perfectly cons
Page 88 and 89: 4. Gauge theories The system of lin
Page 90 and 91: 4. Gauge theories terms are no long
Page 92 and 93: 4. Gauge theories 78
Page 95 and 96: Chapter 5 N = 2 Chern-Simons matter
Page 97 and 98: 5.2. Generalizations representation
Page 99 and 100: 5.2. Generalizations The superpoten
Page 101 and 102: Chapter 6 Quantization, fixed point
Page 103 and 104: 6.1. Quantization of N = 2 Chern-Si
Page 105 and 106: 6.1. Quantization of N = 2 Chern-Si
Page 107 and 108: 6.2. Two-loop renormalization and
Page 109 and 110: 6.2. Two-loop renormalization and
Page 111 and 112: In order to cancel the divergences
Page 113 and 114: 6.3. The spectrum of fixed points W
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Page 119 and 120: 6.4. Infrared stability Figure 6.5:
Page 121 and 122: 6.4. Infrared stability Figure 6.6:
Page 123 and 124: 6.5. A relevant perturbation Figure
Page 125 and 126: 6.5. A relevant perturbation This k
Page 127: Part III Supersymmetry breaking 113
Page 130 and 131: 7. Basics and motivations where the
Page 132 and 133: 7. Basics and motivations It follow
Page 134 and 135: 7. Basics and motivations • the h
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Page 138 and 139: 8. Supersymmetric QCD and Seiberg d
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9. Non-supersymmetric vacua The bou
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9. Non-supersymmetric vacua where t
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9. Non-supersymmetric vacua The phy
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9. Non-supersymmetric vacua conform
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9. Non-supersymmetric vacua the lin
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9. Non-supersymmetric vacua The one
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9. Non-supersymmetric vacua and the
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9. Non-supersymmetric vacua The con
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9. Non-supersymmetric vacua 9.4.4 R
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9. Non-supersymmetric vacua We have
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conclusions 172
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174
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A. Mathematical tools A.2 Useful in
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A. Mathematical tools For SU(N) we
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B. Feynman rules as f = ∇ 2 ∗ V
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B. Feynman rules Matter sector We n
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B. Feynman rules Since terms propor
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B. Feynman rules Φ ∇ α Φ ∇
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B. Feynman rules Here we recognize
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B. Feynman rules and contain an inf
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B. Feynman rules 192 ∂ φ Γ Γ
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C. Details on supersymmetry breakin
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BIBLIOGRAPHY [14] S. Penati and A.
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BIBLIOGRAPHY [47] M. Van Raamsdonk,
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BIBLIOGRAPHY [80] D. Gaiotto and A.
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BIBLIOGRAPHY [114] T. Banks and A.
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